Source code for pyrocko.orthodrome

# http://pyrocko.org - GPLv3
#
# The Pyrocko Developers, 21st Century
# ---|P------/S----------~Lg----------
from __future__ import division, absolute_import

import math
import numpy as num

from .moment_tensor import euler_to_matrix
from .config import config
from .plot.beachball import spoly_cut

from matplotlib.path import Path

d2r = math.pi/180.
r2d = 1./d2r
earth_oblateness = 1./298.257223563
earthradius_equator = 6378.14 * 1000.
earthradius = config().earthradius
d2m = earthradius_equator*math.pi/180.
m2d = 1./d2m

_testpath = Path([(0, 0), (0, 1), (1, 1), (1, 0), (0, 0)], closed=True)

raise_if_slow_path_contains_points = False


[docs]class Slow(Exception): pass
if hasattr(_testpath, 'contains_points') and num.all( _testpath.contains_points([(0.5, 0.5), (1.5, 0.5)]) == [True, False]): def path_contains_points(verts, points): p = Path(verts, closed=True) return p.contains_points(points).astype(num.bool) else: # work around missing contains_points and bug in matplotlib ~ v1.2.0 def path_contains_points(verts, points): if raise_if_slow_path_contains_points: # used by unit test to skip slow gshhg_example.py raise Slow() p = Path(verts, closed=True) result = num.zeros(points.shape[0], dtype=num.bool) for i in range(result.size): result[i] = p.contains_point(points[i, :]) return result try: cbrt = num.cbrt except AttributeError: def cbrt(x): return x**(1./3.) def float_array_broadcast(*args): return num.broadcast_arrays(*[ num.asarray(x, dtype=float) for x in args])
[docs]class Loc(object): ''' Simple location representation. :attrib lat: Latitude in [deg]. :attrib lon: Longitude in [deg]. ''' def __init__(self, lat, lon): self.lat = lat self.lon = lon
[docs]def clip(x, mi, ma): ''' Clip values of an array. :param x: Continunous data to be clipped. :param mi: Clip minimum. :param ma: Clip maximum. :type x: :py:class:`numpy.ndarray` :type mi: float :type ma: float :return: Clipped data. :rtype: :py:class:`numpy.ndarray` ''' return num.minimum(num.maximum(mi, x), ma)
[docs]def wrap(x, mi, ma): ''' Wrapping continuous data to fundamental phase values. .. math:: x_{\\mathrm{wrapped}} = x_{\\mathrm{cont},i} - \\frac{ x_{\\mathrm{cont},i} - r_{\\mathrm{min}} } { r_{\\mathrm{max}} - r_{\\mathrm{min}}} \\cdot ( r_{\\mathrm{max}} - r_{\\mathrm{min}}),\\quad x_{\\mathrm{wrapped}}\\; \\in \\;[ r_{\\mathrm{min}},\\, r_{\\mathrm{max}}]. :param x: Continunous data to be wrapped. :param mi: Minimum value of wrapped data. :param ma: Maximum value of wrapped data. :type x: :py:class:`numpy.ndarray` :type mi: float :type ma: float :return: Wrapped data. :rtype: :py:class:`numpy.ndarray` ''' return x - num.floor((x-mi)/(ma-mi)) * (ma-mi)
def _latlon_pair(args): if len(args) == 2: a, b = args return a.lat, a.lon, b.lat, b.lon elif len(args) == 4: return args
[docs]def cosdelta(*args): ''' Cosine of the angular distance between two points ``a`` and ``b`` on a sphere. This function (find implementation below) returns the cosine of the distance angle 'delta' between two points ``a`` and ``b``, coordinates of which are expected to be given in geographical coordinates and in degrees. For numerical stability a maximum of 1.0 is enforced. .. math:: A_{\\mathrm{lat'}} = \\frac{ \\pi}{180} \\cdot A_{lat}, \\quad A_{\\mathrm{lon'}} = \\frac{ \\pi}{180} \\cdot A_{lon}, \\quad B_{\\mathrm{lat'}} = \\frac{ \\pi}{180} \\cdot B_{lat}, \\quad B_{\\mathrm{lon'}} = \\frac{ \\pi}{180} \\cdot B_{lon}\\\\[0.5cm] \\cos(\\Delta) = \\min( 1.0, \\quad \\sin( A_{\\mathrm{lat'}}) \\sin( B_{\\mathrm{lat'}} ) + \\cos(A_{\\mathrm{lat'}}) \\cos( B_{\\mathrm{lat'}} ) \\cos( B_{\\mathrm{lon'}} - A_{\\mathrm{lon'}} ) :param a: Location point A. :type a: :py:class:`pyrocko.orthodrome.Loc` :param b: Location point B. :type b: :py:class:`pyrocko.orthodrome.Loc` :return: Cosdelta. :rtype: float ''' alat, alon, blat, blon = _latlon_pair(args) return min( 1.0, math.sin(alat*d2r) * math.sin(blat*d2r) + math.cos(alat*d2r) * math.cos(blat*d2r) * math.cos(d2r*(blon-alon)))
[docs]def cosdelta_numpy(a_lats, a_lons, b_lats, b_lons): ''' Cosine of the angular distance between two points ``a`` and ``b`` on a sphere. This function returns the cosines of the distance angles *delta* between two points ``a`` and ``b`` given as :py:class:`numpy.ndarray`. The coordinates are expected to be given in geographical coordinates and in degrees. For numerical stability a maximum of ``1.0`` is enforced. Please find the details of the implementation in the documentation of the function :py:func:`pyrocko.orthodrome.cosdelta` above. :param a_lats: Latitudes in [deg] point A. :param a_lons: Longitudes in [deg] point A. :param b_lats: Latitudes in [deg] point B. :param b_lons: Longitudes in [deg] point B. :type a_lats: :py:class:`numpy.ndarray` :type a_lons: :py:class:`numpy.ndarray` :type b_lats: :py:class:`numpy.ndarray` :type b_lons: :py:class:`numpy.ndarray` :return: Cosdelta. :type b_lons: :py:class:`numpy.ndarray`, ``(N)`` ''' return num.minimum( 1.0, num.sin(a_lats*d2r) * num.sin(b_lats*d2r) + num.cos(a_lats*d2r) * num.cos(b_lats*d2r) * num.cos(d2r*(b_lons-a_lons)))
[docs]def azimuth(*args): ''' Azimuth calculation. This function (find implementation below) returns azimuth ... between points ``a`` and ``b``, coordinates of which are expected to be given in geographical coordinates and in degrees. .. math:: A_{\\mathrm{lat'}} = \\frac{ \\pi}{180} \\cdot A_{lat}, \\quad A_{\\mathrm{lon'}} = \\frac{ \\pi}{180} \\cdot A_{lon}, \\quad B_{\\mathrm{lat'}} = \\frac{ \\pi}{180} \\cdot B_{lat}, \\quad B_{\\mathrm{lon'}} = \\frac{ \\pi}{180} \\cdot B_{lon}\\\\ \\varphi_{\\mathrm{azi},AB} = \\frac{180}{\\pi} \\arctan \\left[ \\frac{ \\cos( A_{\\mathrm{lat'}}) \\cos( B_{\\mathrm{lat'}} ) \\sin(B_{\\mathrm{lon'}} - A_{\\mathrm{lon'}} )} {\\sin ( B_{\\mathrm{lat'}} ) - \\sin( A_{\\mathrm{lat'}} cosdelta) } \\right] :param a: Location point A. :type a: :py:class:`pyrocko.orthodrome.Loc` :param b: Location point B. :type b: :py:class:`pyrocko.orthodrome.Loc` :return: Azimuth in degree ''' alat, alon, blat, blon = _latlon_pair(args) return r2d*math.atan2( math.cos(alat*d2r) * math.cos(blat*d2r) * math.sin(d2r*(blon-alon)), math.sin(d2r*blat) - math.sin(d2r*alat) * cosdelta( alat, alon, blat, blon))
[docs]def azimuth_numpy(a_lats, a_lons, b_lats, b_lons, _cosdelta=None): ''' Calculation of the azimuth (*track angle*) from a location A towards B. This function returns azimuths (*track angles*) from locations A towards B given in :py:class:`numpy.ndarray`. Coordinates are expected to be given in geographical coordinates and in degrees. Please find the details of the implementation in the documentation of the function :py:func:`pyrocko.orthodrome.azimuth`. :param a_lats: Latitudes in [deg] point A. :param a_lons: Longitudes in [deg] point A. :param b_lats: Latitudes in [deg] point B. :param b_lons: Longitudes in [deg] point B. :type a_lats: :py:class:`numpy.ndarray`, ``(N)`` :type a_lons: :py:class:`numpy.ndarray`, ``(N)`` :type b_lats: :py:class:`numpy.ndarray`, ``(N)`` :type b_lons: :py:class:`numpy.ndarray`, ``(N)`` :return: Azimuths in [deg]. :rtype: :py:class:`numpy.ndarray`, ``(N)`` ''' if _cosdelta is None: _cosdelta = cosdelta_numpy(a_lats, a_lons, b_lats, b_lons) return r2d*num.arctan2( num.cos(a_lats*d2r) * num.cos(b_lats*d2r) * num.sin(d2r*(b_lons-a_lons)), num.sin(d2r*b_lats) - num.sin(d2r*a_lats) * _cosdelta)
[docs]def azibazi(*args, **kwargs): ''' Azimuth and backazimuth from location A towards B and back. :returns: Azimuth in [deg] from A to B, back azimuth in [deg] from B to A. :rtype: tuple[float, float] ''' alat, alon, blat, blon = _latlon_pair(args) if alat == blat and alon == blon: return 0., 180. implementation = kwargs.get('implementation', 'c') assert implementation in ('c', 'python') if implementation == 'c': from pyrocko import orthodrome_ext return orthodrome_ext.azibazi(alat, alon, blat, blon) cd = cosdelta(alat, alon, blat, blon) azi = r2d*math.atan2( math.cos(alat*d2r) * math.cos(blat*d2r) * math.sin(d2r*(blon-alon)), math.sin(d2r*blat) - math.sin(d2r*alat) * cd) bazi = r2d*math.atan2( math.cos(blat*d2r) * math.cos(alat*d2r) * math.sin(d2r*(alon-blon)), math.sin(d2r*alat) - math.sin(d2r*blat) * cd) return azi, bazi
[docs]def azibazi_numpy(a_lats, a_lons, b_lats, b_lons, implementation='c'): ''' Azimuth and backazimuth from location A towards B and back. Arguments are given as :py:class:`numpy.ndarray`. :param a_lats: Latitude(s) in [deg] of point A. :type a_lats: :py:class:`numpy.ndarray` :param a_lons: Longitude(s) in [deg] of point A. :type a_lons: :py:class:`numpy.ndarray` :param b_lats: Latitude(s) in [deg] of point B. :type b_lats: :py:class:`numpy.ndarray` :param b_lons: Longitude(s) in [deg] of point B. :type b_lons: :py:class:`numpy.ndarray` :returns: Azimuth(s) in [deg] from A to B, back azimuth(s) in [deg] from B to A. :rtype: :py:class:`numpy.ndarray`, :py:class:`numpy.ndarray` ''' a_lats, a_lons, b_lats, b_lons = float_array_broadcast( a_lats, a_lons, b_lats, b_lons) assert implementation in ('c', 'python') if implementation == 'c': from pyrocko import orthodrome_ext return orthodrome_ext.azibazi_numpy(a_lats, a_lons, b_lats, b_lons) _cosdelta = cosdelta_numpy(a_lats, a_lons, b_lats, b_lons) azis = azimuth_numpy(a_lats, a_lons, b_lats, b_lons, _cosdelta) bazis = azimuth_numpy(b_lats, b_lons, a_lats, a_lons, _cosdelta) eq = num.logical_and(a_lats == b_lats, a_lons == b_lons) ii_eq = num.where(eq)[0] azis[ii_eq] = 0.0 bazis[ii_eq] = 180.0 return azis, bazis
[docs]def azidist_numpy(*args): ''' Calculation of the azimuth (*track angle*) and the distance from locations A towards B on a sphere. The assisting functions used are :py:func:`pyrocko.orthodrome.cosdelta` and :py:func:`pyrocko.orthodrome.azimuth` :param a_lats: Latitudes in [deg] point A. :param a_lons: Longitudes in [deg] point A. :param b_lats: Latitudes in [deg] point B. :param b_lons: Longitudes in [deg] point B. :type a_lats: :py:class:`numpy.ndarray`, ``(N)`` :type a_lons: :py:class:`numpy.ndarray`, ``(N)`` :type b_lats: :py:class:`numpy.ndarray`, ``(N)`` :type b_lons: :py:class:`numpy.ndarray`, ``(N)`` :return: Azimuths in [deg], distances in [deg]. :rtype: :py:class:`numpy.ndarray`, ``(2xN)`` ''' _cosdelta = cosdelta_numpy(*args) _azimuths = azimuth_numpy(_cosdelta=_cosdelta, *args) return _azimuths, r2d*num.arccos(_cosdelta)
[docs]def distance_accurate50m(*args, **kwargs): ''' Accurate distance calculation based on a spheroid of rotation. Function returns distance in meter between points A and B, coordinates of which must be given in geographical coordinates and in degrees. The returned distance should be accurate to 50 m using WGS84. Values for the Earth's equator radius and the Earth's oblateness (``f_oblate``) are defined in the pyrocko configuration file :py:class:`pyrocko.config`. From wikipedia (http://de.wikipedia.org/wiki/Orthodrome), based on: ``Meeus, J.: Astronomical Algorithms, S 85, Willmann-Bell, Richmond 2000 (2nd ed., 2nd printing), ISBN 0-943396-61-1`` .. math:: F = \\frac{\\pi}{180} \\frac{(A_{lat} + B_{lat})}{2}, \\quad G = \\frac{\\pi}{180} \\frac{(A_{lat} - B_{lat})}{2}, \\quad l = \\frac{\\pi}{180} \\frac{(A_{lon} - B_{lon})}{2} \\quad \\\\[0.5cm] S = \\sin^2(G) \\cdot \\cos^2(l) + \\cos^2(F) \\cdot \\sin^2(l), \\quad \\quad C = \\cos^2(G) \\cdot \\cos^2(l) + \\sin^2(F) \\cdot \\sin^2(l) .. math:: w = \\arctan \\left( \\sqrt{ \\frac{S}{C}} \\right) , \\quad r = \\sqrt{\\frac{S}{C} } The spherical-earth distance D between A and B, can be given with: .. math:: D_{sphere} = 2w \\cdot R_{equator} The oblateness of the Earth requires some correction with correction factors h1 and h2: .. math:: h_1 = \\frac{3r - 1}{2C}, \\quad h_2 = \\frac{3r +1 }{2S}\\\\[0.5cm] D = D_{\\mathrm{sphere}} \\cdot [ 1 + h_1 \\,f_{\\mathrm{oblate}} \\cdot \\sin^2(F) \\cos^2(G) - h_2\\, f_{\\mathrm{oblate}} \\cdot \\cos^2(F) \\sin^2(G)] :param a: Location point A. :type a: :py:class:`pyrocko.orthodrome.Loc` :param b: Location point B. :type b: :py:class:`pyrocko.orthodrome.Loc` :return: Distance in [m]. :rtype: float ''' alat, alon, blat, blon = _latlon_pair(args) implementation = kwargs.get('implementation', 'c') assert implementation in ('c', 'python') if implementation == 'c': from pyrocko import orthodrome_ext return orthodrome_ext.distance_accurate50m(alat, alon, blat, blon) f = (alat + blat)*d2r / 2. g = (alat - blat)*d2r / 2. h = (alon - blon)*d2r / 2. s = math.sin(g)**2 * math.cos(h)**2 + math.cos(f)**2 * math.sin(h)**2 c = math.cos(g)**2 * math.cos(h)**2 + math.sin(f)**2 * math.sin(h)**2 w = math.atan(math.sqrt(s/c)) if w == 0.0: return 0.0 r = math.sqrt(s*c)/w d = 2.*w*earthradius_equator h1 = (3.*r-1.)/(2.*c) h2 = (3.*r+1.)/(2.*s) return d * (1. + earth_oblateness * h1 * math.sin(f)**2 * math.cos(g)**2 - earth_oblateness * h2 * math.cos(f)**2 * math.sin(g)**2)
[docs]def distance_accurate50m_numpy( a_lats, a_lons, b_lats, b_lons, implementation='c'): ''' Accurate distance calculation based on a spheroid of rotation. Function returns distance in meter between points ``a`` and ``b``, coordinates of which must be given in geographical coordinates and in degrees. The returned distance should be accurate to 50 m using WGS84. Values for the Earth's equator radius and the Earth's oblateness (``f_oblate``) are defined in the pyrocko configuration file :py:class:`pyrocko.config`. From wikipedia (http://de.wikipedia.org/wiki/Orthodrome), based on: ``Meeus, J.: Astronomical Algorithms, S 85, Willmann-Bell, Richmond 2000 (2nd ed., 2nd printing), ISBN 0-943396-61-1`` .. math:: F_i = \\frac{\\pi}{180} \\frac{(a_{lat,i} + a_{lat,i})}{2}, \\quad G_i = \\frac{\\pi}{180} \\frac{(a_{lat,i} - b_{lat,i})}{2}, \\quad l_i= \\frac{\\pi}{180} \\frac{(a_{lon,i} - b_{lon,i})}{2} \\\\[0.5cm] S_i = \\sin^2(G_i) \\cdot \\cos^2(l_i) + \\cos^2(F_i) \\cdot \\sin^2(l_i), \\quad \\quad C_i = \\cos^2(G_i) \\cdot \\cos^2(l_i) + \\sin^2(F_i) \\cdot \\sin^2(l_i) .. math:: w_i = \\arctan \\left( \\sqrt{\\frac{S_i}{C_i}} \\right), \\quad r_i = \\sqrt{\\frac{S_i}{C_i} } The spherical-earth distance ``D`` between ``a`` and ``b``, can be given with: .. math:: D_{\\mathrm{sphere},i} = 2w_i \\cdot R_{\\mathrm{equator}} The oblateness of the Earth requires some correction with correction factors ``h1`` and ``h2``: .. math:: h_{1.i} = \\frac{3r - 1}{2C_i}, \\quad h_{2,i} = \\frac{3r +1 }{2S_i}\\\\[0.5cm] D_{AB,i} = D_{\\mathrm{sphere},i} \\cdot [1 + h_{1,i} \\,f_{\\mathrm{oblate}} \\cdot \\sin^2(F_i) \\cos^2(G_i) - h_{2,i}\\, f_{\\mathrm{oblate}} \\cdot \\cos^2(F_i) \\sin^2(G_i)] :param a_lats: Latitudes in [deg] point A. :param a_lons: Longitudes in [deg] point A. :param b_lats: Latitudes in [deg] point B. :param b_lons: Longitudes in [deg] point B. :type a_lats: :py:class:`numpy.ndarray`, ``(N)`` :type a_lons: :py:class:`numpy.ndarray`, ``(N)`` :type b_lats: :py:class:`numpy.ndarray`, ``(N)`` :type b_lons: :py:class:`numpy.ndarray`, ``(N)`` :return: Distances in [m]. :rtype: :py:class:`numpy.ndarray`, ``(N)`` ''' a_lats, a_lons, b_lats, b_lons = float_array_broadcast( a_lats, a_lons, b_lats, b_lons) assert implementation in ('c', 'python') if implementation == 'c': from pyrocko import orthodrome_ext return orthodrome_ext.distance_accurate50m_numpy( a_lats, a_lons, b_lats, b_lons) eq = num.logical_and(a_lats == b_lats, a_lons == b_lons) ii_neq = num.where(num.logical_not(eq))[0] if num.all(eq): return num.zeros_like(eq, dtype=float) def extr(x): if isinstance(x, num.ndarray) and x.size > 1: return x[ii_neq] else: return x a_lats = extr(a_lats) a_lons = extr(a_lons) b_lats = extr(b_lats) b_lons = extr(b_lons) f = (a_lats + b_lats)*d2r / 2. g = (a_lats - b_lats)*d2r / 2. h = (a_lons - b_lons)*d2r / 2. s = num.sin(g)**2 * num.cos(h)**2 + num.cos(f)**2 * num.sin(h)**2 c = num.cos(g)**2 * num.cos(h)**2 + num.sin(f)**2 * num.sin(h)**2 w = num.arctan(num.sqrt(s/c)) r = num.sqrt(s*c)/w d = 2.*w*earthradius_equator h1 = (3.*r-1.)/(2.*c) h2 = (3.*r+1.)/(2.*s) dists = num.zeros(eq.size, dtype=float) dists[ii_neq] = d * ( 1. + earth_oblateness * h1 * num.sin(f)**2 * num.cos(g)**2 - earth_oblateness * h2 * num.cos(f)**2 * num.sin(g)**2) return dists
[docs]def ne_to_latlon(lat0, lon0, north_m, east_m): ''' Transform local cartesian coordinates to latitude and longitude. From east and north coordinates (``x`` and ``y`` coordinate :py:class:`numpy.ndarray`) relative to a reference differences in longitude and latitude are calculated, which are effectively changes in azimuth and distance, respectively: .. math:: \\text{distance change:}\\; \\Delta {\\bf{a}} &= \\sqrt{{\\bf{y}}^2 + {\\bf{x}}^2 }/ \\mathrm{R_E}, \\text{azimuth change:}\\; \\Delta \\bf{\\gamma} &= \\arctan( \\bf{x} / \\bf{y}). The projection used preserves the azimuths of the input points. :param lat0: Latitude origin of the cartesian coordinate system in [deg]. :param lon0: Longitude origin of the cartesian coordinate system in [deg]. :param north_m: Northing distances from origin in [m]. :param east_m: Easting distances from origin in [m]. :type north_m: :py:class:`numpy.ndarray`, ``(N)`` :type east_m: :py:class:`numpy.ndarray`, ``(N)`` :type lat0: float :type lon0: float :return: Array with latitudes and longitudes in [deg]. :rtype: :py:class:`numpy.ndarray`, ``(2xN)`` ''' a = num.sqrt(north_m**2+east_m**2)/earthradius gamma = num.arctan2(east_m, north_m) return azidist_to_latlon_rad(lat0, lon0, gamma, a)
[docs]def azidist_to_latlon(lat0, lon0, azimuth_deg, distance_deg): ''' Absolute latitudes and longitudes are calculated from relative changes. Convenience wrapper to :py:func:`azidist_to_latlon_rad` with azimuth and distance given in degrees. :param lat0: Latitude origin of the cartesian coordinate system in [deg]. :type lat0: float :param lon0: Longitude origin of the cartesian coordinate system in [deg]. :type lon0: float :param azimuth_deg: Azimuth from origin in [deg]. :type azimuth_deg: :py:class:`numpy.ndarray`, ``(N)`` :param distance_deg: Distances from origin in [deg]. :type distance_deg: :py:class:`numpy.ndarray`, ``(N)`` :return: Array with latitudes and longitudes in [deg]. :rtype: :py:class:`numpy.ndarray`, ``(2xN)`` ''' return azidist_to_latlon_rad( lat0, lon0, azimuth_deg/180.*num.pi, distance_deg/180.*num.pi)
[docs]def azidist_to_latlon_rad(lat0, lon0, azimuth_rad, distance_rad): ''' Absolute latitudes and longitudes are calculated from relative changes. For numerical stability a range between of ``-1.0`` and ``1.0`` is enforced for ``c`` and ``alpha``. .. math:: \\Delta {\\bf a}_i \\; \\text{and} \\; \\Delta \\gamma_i \\; \\text{are relative distances and azimuths from lat0 and lon0 for \\textit{i} source points of a finite source.} .. math:: \\mathrm{b} &= \\frac{\\pi}{2} -\\frac{\\pi}{180}\\;\\mathrm{lat_0}\\\\ {\\bf c}_i &=\\arccos[\\; \\cos(\\Delta {\\bf{a}}_i) \\cos(\\mathrm{b}) + |\\Delta \\gamma_i| \\, \\sin(\\Delta {\\bf a}_i) \\sin(\\mathrm{b})\\; ] \\\\ \\mathrm{lat}_i &= \\frac{180}{\\pi} \\left(\\frac{\\pi}{2} - {\\bf c}_i \\right) .. math:: \\alpha_i &= \\arcsin \\left[ \\; \\frac{ \\sin(\\Delta {\\bf a}_i ) \\sin(|\\Delta \\gamma_i|)}{\\sin({\\bf c}_i)}\\; \\right] \\\\ \\alpha_i &= \\begin{cases} \\alpha_i, &\\text{if} \\; \\cos(\\Delta {\\bf a}_i) - \\cos(\\mathrm{b}) \\cos({\\bf{c}}_i) > 0, \\; \\text{else} \\\\ \\pi - \\alpha_i, & \\text{if} \\; \\alpha_i > 0,\\; \\text{else}\\\\ -\\pi - \\alpha_i, & \\text{if} \\; \\alpha_i < 0. \\end{cases} \\\\ \\mathrm{lon}_i &= \\mathrm{lon_0} + \\frac{180}{\\pi} \\, \\frac{\\Delta \\gamma_i }{|\\Delta \\gamma_i|} \\cdot \\alpha_i \\text{, with $\\alpha_i \\in [-\\pi,\\pi]$} :param lat0: Latitude origin of the cartesian coordinate system in [deg]. :param lon0: Longitude origin of the cartesian coordinate system in [deg]. :param distance_rad: Distances from origin in [rad]. :param azimuth_rad: Azimuth from origin in [rad]. :type distance_rad: :py:class:`numpy.ndarray`, ``(N)`` :type azimuth_rad: :py:class:`numpy.ndarray`, ``(N)`` :type lat0: float :type lon0: float :return: Array with latitudes and longitudes in [deg]. :rtype: :py:class:`numpy.ndarray`, ``(2xN)`` ''' a = distance_rad gamma = azimuth_rad b = math.pi/2.-lat0*d2r alphasign = 1. alphasign = num.where(gamma < 0, -1., 1.) gamma = num.abs(gamma) c = num.arccos(clip( num.cos(a)*num.cos(b)+num.sin(a)*num.sin(b)*num.cos(gamma), -1., 1.)) alpha = num.arcsin(clip( num.sin(a)*num.sin(gamma)/num.sin(c), -1., 1.)) alpha = num.where( num.cos(a)-num.cos(b)*num.cos(c) < 0, num.where(alpha > 0, math.pi-alpha, -math.pi-alpha), alpha) lat = r2d * (math.pi/2. - c) lon = wrap(lon0 + r2d*alpha*alphasign, -180., 180.) return lat, lon
[docs]def ne_to_latlon_alternative_method(lat0, lon0, north_m, east_m): ''' Transform local cartesian coordinates to latitude and longitude. Like :py:func:`pyrocko.orthodrome.ne_to_latlon`, but this method (implementation below), although it should be numerically more stable, suffers problems at points which are *across the pole* as seen from the cartesian origin. .. math:: \\text{distance change:}\\; \\Delta {{\\bf a}_i} &= \\sqrt{{\\bf{y}}^2_i + {\\bf{x}}^2_i }/ \\mathrm{R_E},\\\\ \\text{azimuth change:}\\; \\Delta {\\bf \\gamma}_i &= \\arctan( {\\bf x}_i {\\bf y}_i). \\\\ \\mathrm{b} &= \\frac{\\pi}{2} -\\frac{\\pi}{180} \\;\\mathrm{lat_0}\\\\ .. math:: {{\\bf z}_1}_i &= \\cos{\\left( \\frac{\\Delta {\\bf a}_i - \\mathrm{b}}{2} \\right)} \\cos {\\left( \\frac{|\\gamma_i|}{2} \\right) }\\\\ {{\\bf n}_1}_i &= \\cos{\\left( \\frac{\\Delta {\\bf a}_i + \\mathrm{b}}{2} \\right)} \\sin {\\left( \\frac{|\\gamma_i|}{2} \\right) }\\\\ {{\\bf z}_2}_i &= \\sin{\\left( \\frac{\\Delta {\\bf a}_i - \\mathrm{b}}{2} \\right)} \\cos {\\left( \\frac{|\\gamma_i|}{2} \\right) }\\\\ {{\\bf n}_2}_i &= \\sin{\\left( \\frac{\\Delta {\\bf a}_i + \\mathrm{b}}{2} \\right)} \\sin {\\left( \\frac{|\\gamma_i|}{2} \\right) }\\\\ {{\\bf t}_1}_i &= \\arctan{\\left( \\frac{{{\\bf z}_1}_i} {{{\\bf n}_1}_i} \\right) }\\\\ {{\\bf t}_2}_i &= \\arctan{\\left( \\frac{{{\\bf z}_2}_i} {{{\\bf n}_2}_i} \\right) } \\\\[0.5cm] c &= \\begin{cases} 2 \\cdot \\arccos \\left( {{\\bf z}_1}_i / \\sin({{\\bf t}_1}_i) \\right),\\; \\text{if } |\\sin({{\\bf t}_1}_i)| > |\\sin({{\\bf t}_2}_i)|,\\; \\text{else} \\\\ 2 \\cdot \\arcsin{\\left( {{\\bf z}_2}_i / \\sin({{\\bf t}_2}_i) \\right)}. \\end{cases}\\\\ .. math:: {\\bf {lat}}_i &= \\frac{180}{ \\pi } \\left( \\frac{\\pi}{2} - {\\bf {c}}_i \\right) \\\\ {\\bf {lon}}_i &= {\\bf {lon}}_0 + \\frac{180}{ \\pi } \\frac{\\gamma_i}{|\\gamma_i|}, \\text{ with}\\; \\gamma \\in [-\\pi,\\pi] :param lat0: Latitude origin of the cartesian coordinate system in [deg]. :param lon0: Longitude origin of the cartesian coordinate system in [deg]. :param north_m: Northing distances from origin in [m]. :param east_m: Easting distances from origin in [m]. :type north_m: :py:class:`numpy.ndarray`, ``(N)`` :type east_m: :py:class:`numpy.ndarray`, ``(N)`` :type lat0: float :type lon0: float :return: Array with latitudes and longitudes in [deg]. :rtype: :py:class:`numpy.ndarray`, ``(2xN)`` ''' b = math.pi/2.-lat0*d2r a = num.sqrt(north_m**2+east_m**2)/earthradius gamma = num.arctan2(east_m, north_m) alphasign = 1. alphasign = num.where(gamma < 0., -1., 1.) gamma = num.abs(gamma) z1 = num.cos((a-b)/2.)*num.cos(gamma/2.) n1 = num.cos((a+b)/2.)*num.sin(gamma/2.) z2 = num.sin((a-b)/2.)*num.cos(gamma/2.) n2 = num.sin((a+b)/2.)*num.sin(gamma/2.) t1 = num.arctan2(z1, n1) t2 = num.arctan2(z2, n2) alpha = t1 + t2 sin_t1 = num.sin(t1) sin_t2 = num.sin(t2) c = num.where( num.abs(sin_t1) > num.abs(sin_t2), num.arccos(z1/sin_t1)*2., num.arcsin(z2/sin_t2)*2.) lat = r2d * (math.pi/2. - c) lon = wrap(lon0 + r2d*alpha*alphasign, -180., 180.) return lat, lon
[docs]def latlon_to_ne(*args): ''' Relative cartesian coordinates with respect to a reference location. For two locations, a reference location A and another location B, given in geographical coordinates in degrees, the corresponding cartesian coordinates are calculated. Assisting functions are :py:func:`pyrocko.orthodrome.azimuth` and :py:func:`pyrocko.orthodrome.distance_accurate50m`. .. math:: D_{AB} &= \\mathrm{distance\\_accurate50m(}A, B \\mathrm{)}, \\quad \\varphi_{\\mathrm{azi},AB} = \\mathrm{azimuth(}A,B \\mathrm{)}\\\\[0.3cm] n &= D_{AB} \\cdot \\cos( \\frac{\\pi }{180} \\varphi_{\\mathrm{azi},AB} )\\\\ e &= D_{AB} \\cdot \\sin( \\frac{\\pi }{180} \\varphi_{\\mathrm{azi},AB}) :param refloc: Location reference point. :type refloc: :py:class:`pyrocko.orthodrome.Loc` :param loc: Location of interest. :type loc: :py:class:`pyrocko.orthodrome.Loc` :return: Northing and easting from refloc to location in [m]. :rtype: tuple[float, float] ''' azi = azimuth(*args) dist = distance_accurate50m(*args) n, e = math.cos(azi*d2r)*dist, math.sin(azi*d2r)*dist return n, e
[docs]def latlon_to_ne_numpy(lat0, lon0, lat, lon): ''' Relative cartesian coordinates with respect to a reference location. For two locations, a reference location (``lat0``, ``lon0``) and another location B, given in geographical coordinates in degrees, the corresponding cartesian coordinates are calculated. Assisting functions are :py:func:`azimuth` and :py:func:`distance_accurate50m`. :param lat0: Latitude of the reference location in [deg]. :param lon0: Longitude of the reference location in [deg]. :param lat: Latitude of the absolute location in [deg]. :param lon: Longitude of the absolute location in [deg]. :return: ``(n, e)``: relative north and east positions in [m]. :rtype: :py:class:`numpy.ndarray`, ``(2xN)`` Implemented formulations: .. math:: D_{AB} &= \\mathrm{distance\\_accurate50m(}A, B \\mathrm{)}, \\quad \\varphi_{\\mathrm{azi},AB} = \\mathrm{azimuth(}A,B \\mathrm{)}\\\\[0.3cm] n &= D_{AB} \\cdot \\cos( \\frac{\\pi }{180} \\varphi_{ \\mathrm{azi},AB} )\\\\ e &= D_{AB} \\cdot \\sin( \\frac{\\pi }{180} \\varphi_{ \\mathrm{azi},AB} ) ''' azi = azimuth_numpy(lat0, lon0, lat, lon) dist = distance_accurate50m_numpy(lat0, lon0, lat, lon) n = num.cos(azi*d2r)*dist e = num.sin(azi*d2r)*dist return n, e
_wgs84 = None def get_wgs84(): global _wgs84 if _wgs84 is None: from geographiclib.geodesic import Geodesic _wgs84 = Geodesic.WGS84 return _wgs84 def amap(n): def wrap(f): if n == 1: def func(*args): it = num.nditer(args + (None,)) for ops in it: ops[-1][...] = f(*ops[:-1]) return it.operands[-1] elif n == 2: def func(*args): it = num.nditer(args + (None, None)) for ops in it: ops[-2][...], ops[-1][...] = f(*ops[:-2]) return it.operands[-2], it.operands[-1] return func return wrap @amap(2) def ne_to_latlon2(lat0, lon0, north_m, east_m): wgs84 = get_wgs84() az = num.arctan2(east_m, north_m)*r2d dist = num.sqrt(east_m**2 + north_m**2) x = wgs84.Direct(lat0, lon0, az, dist) return x['lat2'], x['lon2'] @amap(2) def latlon_to_ne2(lat0, lon0, lat1, lon1): wgs84 = get_wgs84() x = wgs84.Inverse(lat0, lon0, lat1, lon1) dist = x['s12'] az = x['azi1'] n = num.cos(az*d2r)*dist e = num.sin(az*d2r)*dist return n, e @amap(1) def distance_accurate15nm(lat1, lon1, lat2, lon2): wgs84 = get_wgs84() return wgs84.Inverse(lat1, lon1, lat2, lon2)['s12']
[docs]def positive_region(region): ''' Normalize parameterization of a rectangular geographical region. :param region: ``(west, east, south, north)`` in [deg]. :type region: tuple of float :returns: ``(west, east, south, north)``, where ``west <= east`` and where ``west`` and ``east`` are in the range ``[-180., 180.+360.]``. :rtype: tuple of float ''' west, east, south, north = [float(x) for x in region] assert -180. - 360. <= west < 180. assert -180. < east <= 180. + 360. assert -90. <= south < 90. assert -90. < north <= 90. if east < west: east += 360. if west < -180.: west += 360. east += 360. return (west, east, south, north)
[docs]def points_in_region(p, region): ''' Check what points are contained in a rectangular geographical region. :param p: ``(lat, lon)`` pairs in [deg]. :type p: :py:class:`numpy.ndarray` ``(N, 2)`` :param region: ``(west, east, south, north)`` region boundaries in [deg]. :type region: tuple of float :returns: Mask, returning ``True`` for each point within the region. :rtype: :py:class:`numpy.ndarray` of bool, shape ``(N)`` ''' w, e, s, n = positive_region(region) return num.logical_and( num.logical_and(s <= p[:, 0], p[:, 0] <= n), num.logical_or( num.logical_and(w <= p[:, 1], p[:, 1] <= e), num.logical_and(w-360. <= p[:, 1], p[:, 1] <= e-360.)))
[docs]def point_in_region(p, region): ''' Check if a point is contained in a rectangular geographical region. :param p: ``(lat, lon)`` in [deg]. :type p: tuple of float :param region: ``(west, east, south, north)`` region boundaries in [deg]. :type region: tuple of float :returns: ``True``, if point is in region, else ``False``. :rtype: bool ''' w, e, s, n = positive_region(region) return num.logical_and( num.logical_and(s <= p[0], p[0] <= n), num.logical_or( num.logical_and(w <= p[1], p[1] <= e), num.logical_and(w-360. <= p[1], p[1] <= e-360.)))
[docs]def radius_to_region(lat, lon, radius): ''' Get a rectangular region which fully contains a given circular region. :param lat: Latitude of the center point of circular region in [deg]. :type lat: float :param lon: Longitude of the center point of circular region in [deg]. :type lon: float :param radius: Radius of circular region in [m]. :type radius: float :returns: Rectangular region as ``(east, west, south, north)`` in [deg] or ``None``. :rtype: tuple[float, float, float, float] ''' radius_deg = radius * m2d if radius_deg < 45.: lat_min = max(-90., lat - radius_deg) lat_max = min(90., lat + radius_deg) absmaxlat = max(abs(lat_min), abs(lat_max)) if absmaxlat > 89: lon_min = -180. lon_max = 180. else: lon_min = max( -180. - 360., lon - radius_deg / math.cos(absmaxlat*d2r)) lon_max = min( 180. + 360., lon + radius_deg / math.cos(absmaxlat*d2r)) lon_min, lon_max, lat_min, lat_max = positive_region( (lon_min, lon_max, lat_min, lat_max)) return lon_min, lon_max, lat_min, lat_max else: return None
[docs]def geographic_midpoint(lats, lons, weights=None): ''' Calculate geographic midpoints by finding the center of gravity. This method suffers from instabilities if points are centered around the poles. :param lats: Latitudes in [deg]. :param lons: Longitudes in [deg]. :param weights: Weighting factors. :type lats: :py:class:`numpy.ndarray`, ``(N)`` :type lons: :py:class:`numpy.ndarray`, ``(N)`` :type weights: optional, :py:class:`numpy.ndarray`, ``(N)`` :return: Latitudes and longitudes of the midpoints in [deg]. :rtype: :py:class:`numpy.ndarray`, ``(2xN)`` ''' if not weights: weights = num.ones(len(lats)) total_weigth = num.sum(weights) weights /= total_weigth lats = lats * d2r lons = lons * d2r x = num.sum(num.cos(lats) * num.cos(lons) * weights) y = num.sum(num.cos(lats) * num.sin(lons) * weights) z = num.sum(num.sin(lats) * weights) lon = num.arctan2(y, x) hyp = num.sqrt(x**2 + y**2) lat = num.arctan2(z, hyp) return lat/d2r, lon/d2r
def geographic_midpoint_locations(locations, weights=None): coords = num.array([loc.effective_latlon for loc in locations]) return geographic_midpoint(coords[:, 0], coords[:, 1], weights)
[docs]def geodetic_to_ecef(lat, lon, alt): ''' Convert geodetic coordinates to Earth-Centered, Earth-Fixed (ECEF) Cartesian coordinates. [#1]_ [#2]_ :param lat: Geodetic latitude in [deg]. :param lon: Geodetic longitude in [deg]. :param alt: Geodetic altitude (height) in [m] (positive for points outside the geoid). :type lat: float :type lon: float :type alt: float :return: ECEF Cartesian coordinates (X, Y, Z) in [m]. :rtype: tuple[float, float, float] .. [#1] https://en.wikipedia.org/wiki/ECEF .. [#2] https://en.wikipedia.org/wiki/Geographic_coordinate_conversion #From_geodetic_to_ECEF_coordinates ''' f = earth_oblateness a = earthradius_equator e2 = 2*f - f**2 lat, lon = num.radians(lat), num.radians(lon) # Normal (plumb line) N = a / num.sqrt(1.0 - (e2 * num.sin(lat)**2)) X = (N+alt) * num.cos(lat) * num.cos(lon) Y = (N+alt) * num.cos(lat) * num.sin(lon) Z = (N*(1.0-e2) + alt) * num.sin(lat) return (X, Y, Z)
[docs]def ecef_to_geodetic(X, Y, Z): ''' Convert Earth-Centered, Earth-Fixed (ECEF) Cartesian coordinates to geodetic coordinates (Ferrari's solution). :param X, Y, Z: Cartesian coordinates in ECEF system in [m]. :type X, Y, Z: float :return: Geodetic coordinates (lat, lon, alt). Latitude and longitude are in [deg] and altitude is in [m] (positive for points outside the geoid). :rtype: tuple, float .. seealso :: https://en.wikipedia.org/wiki/Geographic_coordinate_conversion #The_application_of_Ferrari.27s_solution ''' f = earth_oblateness a = earthradius_equator b = a * (1. - f) e2 = 2.*f - f**2 # usefull a2 = a**2 b2 = b**2 e4 = e2**2 X2 = X**2 Y2 = Y**2 Z2 = Z**2 r = num.sqrt(X2 + Y2) r2 = r**2 e_prime2 = (a2 - b2)/b2 E2 = a2 - b2 F = 54. * b2 * Z2 G = r2 + (1.-e2)*Z2 - (e2*E2) C = (e4 * F * r2) / (G**3) S = cbrt(1. + C + num.sqrt(C**2 + 2.*C)) P = F / (3. * (S + 1./S + 1.)**2 * G**2) Q = num.sqrt(1. + (2.*e4*P)) dum1 = -(P*e2*r) / (1.+Q) dum2 = 0.5 * a2 * (1. + 1./Q) dum3 = (P * (1.-e2) * Z2) / (Q * (1.+Q)) dum4 = 0.5 * P * r2 r0 = dum1 + num.sqrt(dum2 - dum3 - dum4) U = num.sqrt((r - e2*r0)**2 + Z2) V = num.sqrt((r - e2*r0)**2 + (1.-e2)*Z2) Z0 = (b2*Z) / (a*V) alt = U * (1. - (b2 / (a*V))) lat = num.arctan((Z + e_prime2 * Z0)/r) lon = num.arctan2(Y, X) return (lat*r2d, lon*r2d, alt)
[docs]class Farside(Exception): pass
def latlon_to_xyz(latlons): if latlons.ndim == 1: return latlon_to_xyz(latlons[num.newaxis, :])[0] points = num.zeros((latlons.shape[0], 3)) lats = latlons[:, 0] lons = latlons[:, 1] points[:, 0] = num.cos(lats*d2r) * num.cos(lons*d2r) points[:, 1] = num.cos(lats*d2r) * num.sin(lons*d2r) points[:, 2] = num.sin(lats*d2r) return points def xyz_to_latlon(xyz): if xyz.ndim == 1: return xyz_to_latlon(xyz[num.newaxis, :])[0] latlons = num.zeros((xyz.shape[0], 2)) latlons[:, 0] = num.arctan2( xyz[:, 2], num.sqrt(xyz[:, 0]**2 + xyz[:, 1]**2)) * r2d latlons[:, 1] = num.arctan2( xyz[:, 1], xyz[:, 0]) * r2d return latlons def rot_to_00(lat, lon): rot0 = euler_to_matrix(0., -90.*d2r, 0.).A rot1 = euler_to_matrix(-d2r*lat, 0., -d2r*lon).A return num.dot(rot0.T, num.dot(rot1, rot0)).T def distances3d(a, b): return num.sqrt(num.sum((a-b)**2, axis=a.ndim-1)) def circulation(points2): return num.sum( (points2[1:, 0] - points2[:-1, 0]) * (points2[1:, 1] + points2[:-1, 1])) def stereographic(points): dists = distances3d(points[1:, :], points[:-1, :]) if dists.size > 0: maxdist = num.max(dists) cutoff = maxdist**2 / 2. else: cutoff = 1.0e-5 points = points.copy() if num.any(points[:, 0] < -1. + cutoff): raise Farside() points_out = points[:, 1:].copy() factor = 1.0 / (1.0 + points[:, 0]) points_out *= factor[:, num.newaxis] return points_out def stereographic_poly(points): dists = distances3d(points[1:, :], points[:-1, :]) if dists.size > 0: maxdist = num.max(dists) cutoff = maxdist**2 / 2. else: cutoff = 1.0e-5 points = points.copy() if num.any(points[:, 0] < -1. + cutoff): raise Farside() points_out = points[:, 1:].copy() factor = 1.0 / (1.0 + points[:, 0]) points_out *= factor[:, num.newaxis] if circulation(points_out) >= 0: raise Farside() return points_out def gnomonic_x(points, cutoff=0.01): points_out = points[:, 1:].copy() if num.any(points[:, 0] < cutoff): raise Farside() factor = 1.0 / points[:, 0] points_out *= factor[:, num.newaxis] return points_out def cneg(i, x): if i == 1: return x else: return num.logical_not(x)
[docs]def contains_points(polygon, points): ''' Test which points are inside polygon on a sphere. The inside of the polygon is defined as the area which is to the left hand side of an observer walking the polygon line, points in order, on the sphere. Lines between the polygon points are treated as great circle paths. The polygon may be arbitrarily complex, as long as it does not have any crossings or thin parts with zero width. The polygon may contain the poles and is allowed to wrap around the sphere multiple times. The algorithm works by consecutive cutting of the polygon into (almost) hemispheres and subsequent Gnomonic projections to perform the point-in-polygon tests on a 2D plane. :param polygon: Point coordinates defining the polygon [deg]. :type polygon: :py:class:`numpy.ndarray` of shape ``(N, 2)``, second index 0=lat, 1=lon :param points: Coordinates of points to test [deg]. :type points: :py:class:`numpy.ndarray` of shape ``(N, 2)``, second index 0=lat, 1=lon :returns: Boolean mask array. :rtype: :py:class:`numpy.ndarray` of shape ``(N,)``. ''' and_ = num.logical_and points_xyz = latlon_to_xyz(points) mask_x = 0. <= points_xyz[:, 0] mask_y = 0. <= points_xyz[:, 1] mask_z = 0. <= points_xyz[:, 2] result = num.zeros(points.shape[0], dtype=int) for ix in [-1, 1]: for iy in [-1, 1]: for iz in [-1, 1]: mask = and_( and_(cneg(ix, mask_x), cneg(iy, mask_y)), cneg(iz, mask_z)) center_xyz = num.array([ix, iy, iz], dtype=float) lat, lon = xyz_to_latlon(center_xyz) rot = rot_to_00(lat, lon) points_rot_xyz = num.dot(rot, points_xyz[mask, :].T).T points_rot_pro = gnomonic_x(points_rot_xyz) offset = 0.01 poly_xyz = latlon_to_xyz(polygon) poly_rot_xyz = num.dot(rot, poly_xyz.T).T poly_rot_xyz[:, 0] -= offset groups = spoly_cut([poly_rot_xyz], axis=0) for poly_rot_group_xyz in groups[1]: poly_rot_group_xyz[:, 0] += offset poly_rot_group_pro = gnomonic_x( poly_rot_group_xyz) if circulation(poly_rot_group_pro) > 0: result[mask] += path_contains_points( poly_rot_group_pro, points_rot_pro) else: result[mask] -= path_contains_points( poly_rot_group_pro, points_rot_pro) return result.astype(num.bool)
[docs]def contains_point(polygon, point): ''' Test if point is inside polygon on a sphere. Convenience wrapper to :py:func:`contains_points` to test a single point. :param polygon: Point coordinates defining the polygon [deg]. :type polygon: :py:class:`numpy.ndarray` of shape ``(N, 2)``, second index 0=lat, 1=lon :param point: Coordinates ``(lat, lon)`` of point to test [deg]. :type point: tuple of float :returns: ``True``, if point is located within polygon, else ``False``. :rtype: bool ''' return bool( contains_points(polygon, num.asarray(point)[num.newaxis, :])[0])